Texture Painting not working with a blank image
Your issue is the blend mode you're using. When you paint with the add blend mode, it doesn't effect the alpha channel at all. So if you paint on transparency, it's still transparent. When you paint on a part that partly transparent, it only effects the color, and the opacity stays the same.

The solution is to instead use the mix blend mode. Instead of adding to the values of the image like the add blend mode does, this mixes the two values.